Q. 28. If in an A.P. the sum of m terms is equal to n andthe sum of n

1.	Q. 28. If in an A.P. the sum of m terms is equal to n andthe sum of n terms is equal to m, then prove thatthe sum of (m +n) terms is -(mn).
Answer» First term = a , common difference = d Sm = n →m/2(2a + (m-1)d) =n …….(1) Sn = m →n/2(2a + (n-1)d) =m ……..(2) Eq. (2) – (1) 2a(n-m) + d(n2-n-m2+m)=2m-n n-m{2a + d(n+m-1)}=-2(n-m) …….(3) 2a + d(n+m-1) = -2 Now, Sm+n = m+n/2{2a+(m+n-1)d} =-(m+n) (usingeq. 3)

Q. 28. If in an A.P. the sum of m terms is equal to n andthe sum of n terms is equal to m, then prove thatthe sum of (m +n) terms is -(mn).

Answer»

First term = a , common difference = d

Sm = n

→m/2(2a + (m-1)d) =n …….(1)

Sn = m

→n/2(2a + (n-1)d) =m ……..(2)

Eq. (2) – (1)

2a(n-m) + d(n2-n-m2+m)=2m-n

n-m{2a + d(n+m-1)}=-2(n-m) …….(3)

2a + d(n+m-1) = -2

Now, Sm+n = m+n/2{2a+(m+n-1)d}

=-(m+n) (usingeq. 3)